scholarly journals Central Limit Theorem in Multitype Branching Random Walk

2009 ◽  
Vol 5 (2) ◽  
pp. 207-220
Author(s):  
A. Rahimzadeh Sani ◽  
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Branching Random Walk

scholarly journals A central limit theorem and a law of the iterated logarithm for the Biggins martingale of the supercritical branching random walk

2016 ◽  
Vol 53 (4) ◽  
pp. 1178-1192 ◽  
Author(s):  
Alexander Iksanov ◽  
Zakhar Kabluchko
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Functional Central Limit Theorem ◽  
Branching Random Walk ◽  
Iterated Logarithm ◽  
Functional Central Limit

Abstract Let (Wn(θ))n∈ℕ0 be the Biggins martingale associated with a supercritical branching random walk, and denote by W_∞(θ) its limit. Assuming essentially that the martingale (Wn(2θ))n∈ℕ0 is uniformly integrable and that var W1(θ) is finite, we prove a functional central limit theorem for the tail process (W∞(θ)-Wn+r(θ))r∈ℕ0 and a law of the iterated logarithm for W∞(θ)-Wn(θ) as n→∞.

Heavy traffic limit theorems in fluctuation theory

1978 ◽  
Vol 10 (04) ◽  
pp. 852-866
Author(s):  
A. J. Stam
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
The Central Limit Theorem ◽  
Heavy Traffic Limit ◽  
Oscillating Random Walk

Let be a family of random walks with For ε↓0 under certain conditions the random walk U (∊) n converges to an oscillating random walk. The ladder point distributions and expectations converge correspondingly. Let M ∊ = max {U (∊) n , n ≧ 0}, v 0 = min {n : U (∊) n = M ∊}, v 1 = max {n : U (∊) n = M ∊}. The joint limiting distribution of ∊2σ∊ –2 v 0 and ∊σ∊ –2 M ∊ is determined. It is the same as for ∊2σ∊ –2 v 1 and ∊σ–2 ∊ M ∊. The marginal ∊σ–2 ∊ M ∊ gives Kingman's heavy traffic theorem. Also lim ∊–1 P(M ∊ = 0) and lim ∊–1 P(M ∊ < x) are determined. Proofs are by direct comparison of corresponding probabilities for U (∊) n and for a special family of random walks related to MI/M/1 queues, using the central limit theorem.

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